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Let q≥5 be a prime number. Let be a quadratic number field, where . Then the class number of k is divisible by q for certain integers u, w. Conversely, assume Ω/k is an unramified cyclic extension of degree q(which implies the class number of k is divisible by q), and Ω is the splitting field of some irreducible trinomial f(X) = Xq-aX-b with integer coefficients, with D(f) the discriminant of f(X). Then f(X) must be of the form f(X= Xq-uq−2 wX-uq−1 in a certain sense where u, w are certain integers. Therefore, with . Moreover, the above two results are both generalized for certain kinds of general polynomials.